Optimized aerodynamic profile for a turbine vane, in particular for a nozzle of the fifth stage of a turbine

ABSTRACT

When cold and in the non-coated state, the aerodynamic profile is substantially identical to a nominal profile determined by the Cartesian coordinates X,Y, Zadim given in Table 1, in which the coordinate Zadim is the quotient D/H where D is the distance of the point under consideration from a first reference plane P 0  situated at the base of the nominal profile, and H is the height of said profile measured from the first reference plane to a second reference plane P 1 . The measurements D and H are taken radially relative to the axis of the turbine, while the X coordinate is measured in the axial direction of the turbine.

CROSS-REFERENCE TO RELATED APPLICATION(S)

This patent application claims the benefit under 35 U.S.C. § 119 to U.S. Provisional Application No. 62/361,551, filed on Jul. 13, 2016, the entirety of which is incorporated herein by reference.

FIELD OF THE DISCLOSURE

The present invention relates to an aerodynamic profile for a turbine vane.

Introduction

In particular, the invention relates to the aerodynamic profile of a nozzle vane forming a portion of the stator of a gas turbine, and more particularly of a low pressure turbine of the type used in a turbojet of an aircraft.

The invention relates in particular to a nozzle vane of the fifth stage of a turbine having a plurality of stages, preferably seven stages.

Such a profile should enable the turbine to provide the desired efficiency, and in order to do that it must be such that the flow of air around the profile is sound over the working spectrum of the turbine, i.e. substantially such that it does not give rise to turbulence, which is harmful for overall efficiency.

It must be capable of being installed properly in the environment of the engine, and in particular, for a nozzle vane forming a portion of the stator of the turbine, it must be capable of being fastened easily to the sectors of the foot platform (the zone of the vane that is furthest from the axis of rotation of the turbine) and of the head platform (the zone that is closest to said axis). Furthermore, this part must present a profile enabling it to be manufactured in a reliable and cost-effective manner by available manufacturing methods, such as casting, forging, machining, additive fabrication, or else weaving, without this list being limitative.

Furthermore, the profile of the vane must enable it to withstand the mechanical stresses, to which the vane is subjected, by enabling those stresses to be spread over the entire vane in such a manner as to avoid premature wear thereof. This spreading must apply both in static mode and in dynamic mode.

An object of the invention is to propose an aerodynamic profile for a turbine vane that is optimized, and capable of satisfying those objectives.

This object is achieved by the fact that when cold and in a non-coated state, said profile is substantially identical to a nominal profile determined by the Cartesian coordinates X,Y,Zadim given in Table 1, in which the coordinate Zadim is the quotient D/H where D is the distance of the point P under consideration from a reference X,Y plane situated at the base of the nominal profile, and H is the height of said profile measured from said reference plane that is the intersection of the stacking axis of the set of vanes and the axisymmetric surface of the hub, out to a second reference plane that is the intersection of said stacking axis with the axisymmetric surface of the casing, the measurements D and H being taken radially relative to the axis of the turbine, while the coordinate X is measured in the axial direction of the turbine.

This profile has been determined as a result of numerous tests and simulations. It is defined cold, i.e. at an ambient temperature of 20° C. That is a reference temperature at which the profile is geometrically determined. The above-mentioned objectives of aerodynamics and mechanical optimization are naturally valid for the conditions of use of this aerodynamic profile, i.e. when hot, at a temperature that is stabilized when the engine of which the turbine forms a part is being used under cruising conditions.

Furthermore, the aerodynamic profile of the invention is defined in the non-coated state. Since turbine vanes are subjected to high temperature gradients, it is common practice for them to be provided with a coating having thermal properties enabling them more easily to withstand such temperature variations. The profile is determined prior to installing such a coating.

It is stated above that the profile of the invention is “substantially identical” to the nominal profile. This means that the profile may depart very slightly from said nominal profile.

The aerodynamic profile is thus preferably defined within an envelope of ±1 millimeter (mm) in a direction normal to the surface of the nominal profile.

This variation takes account in particular of manufacturing tolerances of the profile.

It is also preferable for the X,Y coordinates of the aerodynamic profile to lie within a range of ±5% relative to the X,Y coordinates of the nominal profile.

This variation takes account of the setting of the profile to adapt to the flow coming from the blades of the turbine situated upstream, so as to further improve the efficiency of the turbine. In particular, the profile is adapted so as to be optimized for the flow coming from the rotary wheel of the fourth stage of a turbine when the vane of the invention belongs to the nozzle of the fifth stage of a turbine.

With a nozzle vane, i.e. a portion of the stator of a turbine, the profile serves to orient the fluid correctly for the turbine blades that are situated downstream, and that belong in particular to the rotary wheel of the fifth stage of the turbine.

BRIEF DESCRIPTION OF THE DRAWINGS

The invention can be better understood and its advantages appear better on reading the following detailed description of an embodiment given by way of non-limiting example. The description refers to the accompanying drawings, in which:

FIG. 1 is a fragmentary axial section view of a turbine including a vane of profile that corresponds to that of the invention;

FIG. 2 is an enlarged view of FIG. 1, showing the fifth stage of the turbine.

DETAILED DESCRIPTION

The turbine shown in axial section in FIG. 1 comprises seven stages referenced respectively S1 to S7. In the direction DF going from upstream to downstream, each stage comprises a nozzle that forms a portion of the stator of the turbine and that has a plurality of radially-oriented vanes, and a rotary wheel that forms a portion of the rotor of the turbine and that likewise includes a plurality of radially-oriented blades.

In FIG. 1, the vanes of the nozzles of stages S1 to S7 are given references N1 to N7, whereas the blades of rotary wheels of stages S1 to S7 are given respective references W1 to W7.

In known manner, the vanes of the nozzles are fastened at both ends to stationary structure portions, while the blades of the rotary wheels are fastened to a rotary disk via their roots that are formed at their radially-inner ends closer to the axis of the rotation A of the turbine. The disks d1 to d7 forming parts of the wheels W1 to W7 are constrained to rotate together.

The invention relates in particular to a vane N5 which is a vane of the nozzle of the fifth stage S5 of the turbine that, as mentioned above, preferably comprises seven stages as shown, without this number being limiting.

As best seen in FIG. 2, via its foot 10, which is its end remote from the axis of rotation A, the vane N5 is fastened to the outer ring 12 of the turbine, which ring is stationary in rotation. Via its head 14, formed by its end closer to the axis of rotation A, the vane is fastened to an inner shroud 16 of the turbine. By convention, the aerodynamic profile 18 of the vane is the entire portion of said vane that extends radially outwards from its head 14 to its foot 10 without incorporating the fastenings respectively to the outer ring 12 and to the inner shroud 16.

In FIG. 2, there can be seen a frame of reference comprising Cartesian coordinates X,Y,Z. The radial direction Z is the height direction of the vane, which extends radially from its head to its foot. This direction Z is perpendicular to the axial direction X which is the direction of the axis of rotation A of the turbine. The direction Y is perpendicular to the X,Z plane and is therefore tangential to the direction of rotation of the turbine.

The nominal profile from which the aerodynamic profile of the invention is determined is defined in following Table 1 of coordinates, in which the coordinate Zadim, measured along the axis Z is non-dimensional, whereas the dimensions X and Y, respectively measured along the axes X and Y, are expressed in millimeters.

TABLE 1 X Y Zadim −13.4934 2.846353 −0.05624 −13.5334 2.808071 −0.05624 −13.5807 2.743066 −0.05624 −13.6257 2.627408 −0.05624 −13.6403 2.454967 −0.05624 −13.6038 2.232309 −0.05624 −13.5103 1.966179 −0.05624 −13.357 1.662348 −0.05624 −13.1409 1.325613 −0.05624 −12.8586 0.961799 −0.05624 −12.5054 0.579086 −0.05624 −12.0762 0.188494 −0.05624 −11.567 −0.19641 −0.05624 −10.9758 −0.56014 −0.05624 −10.3032 −0.8862 −0.05624 −9.5534 −1.15791 −0.05624 −8.73376 −1.35896 −0.05624 −7.85507 −1.4732 −0.05624 −6.93147 −1.48534 −0.05624 −5.97941 −1.37996 −0.05624 −5.0221 −1.14649 −0.05624 −4.0826 −0.781 −0.05624 −3.18218 −0.29065 −0.05624 −2.33541 0.30782 −0.05624 −1.54972 0.99362 −0.05624 −0.82859 1.74734 −0.05624 −0.16963 2.548846 −0.05624 0.433936 3.379088 −0.05624 0.98771 4.22331 −0.05624 1.494313 5.070795 −0.05624 1.961746 5.909104 −0.05624 2.394726 6.729038 −0.05624 2.794595 7.524035 −0.05624 3.166529 8.286277 −0.05624 3.512166 9.010361 −0.05624 3.832253 9.691878 −0.05624 4.129144 10.32624 −0.05624 4.402253 10.91082 −0.05624 4.651346 11.44348 −0.05624 4.877349 11.92218 −0.05624 5.079241 12.34656 −0.05624 5.257046 12.71652 −0.05624 5.410652 13.03288 −0.05624 5.540215 13.29722 −0.05624 5.646365 13.51184 −0.05624 5.730012 13.67994 −0.05624 5.755476 13.81786 −0.05624 5.723704 13.91065 −0.05624 5.685173 13.9589 −0.05624 5.647365 13.98754 −0.05624 5.609918 14.00534 −0.05624 5.55723 14.01704 −0.05624 5.472188 14.00705 −0.05624 5.371835 13.93688 −0.05624 5.286174 13.79694 −0.05624 5.179784 13.61672 −0.05624 5.049129 13.39505 −0.05624 4.893271 13.1301 −0.05624 4.711606 12.82072 −0.05624 4.503815 12.46638 −0.05624 4.269674 12.06729 −0.05624 4.009438 11.62408 −0.05624 3.723078 11.13821 −0.05624 3.411203 10.61142 −0.05624 3.073319 10.04666 −0.05624 2.710092 9.446749 −0.05624 2.321131 8.815655 −0.05624 1.905549 8.158152 −0.05624 1.463931 7.478571 −0.05624 0.993735 6.783767 −0.05624 0.494626 6.07969 −0.05624 −0.0344 5.373329 −0.05624 −0.59741 4.674749 −0.05624 −1.19467 3.992152 −0.05624 −1.82781 3.336021 −0.05624 −2.49992 2.720176 −0.05624 −3.21178 2.159178 −0.05624 −3.9605 1.666842 −0.05624 −4.74174 1.259704 −0.05624 −5.54666 0.953693 −0.05624 −6.36053 0.759601 −0.05624 −7.16509 0.679321 −0.05624 −7.9384 0.705753 −0.05624 −8.66684 0.822823 −0.05624 −9.33847 1.009754 −0.05624 −9.9485 1.243705 −0.05624 −10.4973 1.50251 −0.05624 −10.9883 1.767547 −0.05624 −11.4256 2.024523 −0.05624 −11.8131 2.263363 −0.05624 −12.1555 2.475077 −0.05624 −12.456 2.653824 −0.05624 −12.7185 2.793617 −0.05624 −12.9453 2.889481 −0.05624 −13.1371 2.936717 −0.05624 −13.2882 2.937457 −0.05624 −13.3927 2.908525 −0.05624 −13.6519 2.964973 −0.02812 −13.6925 2.926345 −0.02812 −13.7407 2.86067 −0.02812 −13.787 2.743791 −0.02812 −13.8036 2.569272 −0.02812 −13.7704 2.343117 −0.02812 −13.681 2.071731 −0.02812 −13.5322 1.760707 −0.02812 −13.321 1.414794 −0.02812 −13.0435 1.039774 −0.02812 −12.6949 0.643698 −0.02812 −12.27 0.237395 −0.02812 −11.7644 −0.16571 −0.02812 −11.1755 −0.55004 −0.02812 −10.5033 −0.89858 −0.02812 −9.75108 −1.19349 −0.02812 −8.9257 −1.41648 −0.02812 −8.03779 −1.54894 −0.02812 −7.10246 −1.57305 −0.02812 −6.13728 −1.47203 −0.02812 −5.1675 −1.23541 −0.02812 −4.21744 −0.86086 −0.02812 −3.3089 −0.35753 −0.02812 −2.45647 0.256297 −0.02812 −1.66725 0.958686 −0.02812 −0.94417 1.729313 −0.02812 −0.28429 2.547554 −0.02812 0.319387 3.394239 −0.02812 0.872654 4.254517 −0.02812 1.378417 5.117486 −0.02812 1.844789 5.970702 −0.02812 2.276627 6.8049 −0.02812 2.675326 7.613464 −0.02812 3.046101 8.388569 −0.02812 3.390713 9.124728 −0.02812 3.70989 9.817515 −0.02812 4.006016 10.46231 −0.02812 4.278461 11.05649 −0.02812 4.527094 11.59783 −0.02812 4.752722 12.08434 −0.02812 4.954376 12.51562 −0.02812 5.132026 12.89159 −0.02812 5.28555 13.21309 −0.02812 5.415085 13.48172 −0.02812 5.521236 13.69982 −0.02812 5.604913 13.87063 −0.02812 5.631656 14.01016 −0.02812 5.59951 14.10416 −0.02812 5.56017 14.15281 −0.02812 5.521563 14.18141 −0.02812 5.483414 14.19889 −0.02812 5.4299 14.20982 −0.02812 5.344049 14.19777 −0.02812 5.244616 14.12393 −0.02812 5.158928 13.98169 −0.02812 5.052293 13.79868 −0.02812 4.921377 13.57356 −0.02812 4.765255 13.30447 −0.02812 4.58334 12.99022 −0.02812 4.375352 12.63027 −0.02812 4.141104 12.22477 −0.02812 3.880865 11.77438 −0.02812 3.594683 11.28053 −0.02812 3.283139 10.74497 −0.02812 2.945812 10.17065 −0.02812 2.583414 9.560369 −0.02812 2.1955 8.918152 −0.02812 1.781256 8.248737 −0.02812 1.341216 7.556515 −0.02812 0.872756 6.848364 −0.02812 0.375529 6.130249 −0.02812 −0.1516 5.409228 −0.02812 −0.71285 4.695412 −0.02812 −1.30875 3.997165 −0.02812 −1.94116 3.325125 −0.02812 −2.61346 2.693327 −0.02812 −3.32691 2.116799 −0.02812 −4.07913 1.610213 −0.02812 −4.8661 1.191167 −0.02812 −5.67896 0.87695 −0.02812 −6.50252 0.680065 −0.02812 −7.31733 0.603798 −0.02812 −8.09945 0.641131 −0.02812 −8.83427 0.774419 −0.02812 −9.50909 0.980442 −0.02812 −10.1197 1.234245 −0.02812 −10.6672 1.512406 −0.02812 −11.1557 1.79568 −0.02812 −11.5899 2.069499 −0.02812 −11.9741 2.323678 −0.02812 −12.3135 2.549064 −0.02812 −12.6115 2.739982 −0.02812 −12.8722 2.890405 −0.02812 −13.0983 2.99536 −0.02812 −13.2905 3.04985 −0.02812 −13.4434 3.054889 −0.02812 −13.5496 3.02725 −0.02812 −13.8103 3.083561 0 −13.8516 3.044581 0 −13.9007 2.978225 0 −13.9483 2.860103 0 −13.967 2.683501 0 −13.9369 2.453865 0 −13.8516 2.177267 0 −13.7075 1.859117 0 −13.5012 1.504114 0 −13.2287 1.117993 0 −12.8849 0.708673 0 −12.4646 0.286796 0 −11.9629 −0.13436 0 −11.3766 −0.53915 0 −10.7051 −0.91005 0 −9.95095 −1.22808 0 −9.12019 −1.47309 0 −8.22326 −1.62402 0 −7.27617 −1.66056 0 −6.29772 −1.56447 0 −5.31517 −1.32524 0 −4.35421 −0.94205 0 −3.43726 −0.42601 0 −2.57892 0.203001 0 −1.786 0.9219 0 −1.06081 1.709415 0 −0.39991 2.54441 0 0.203984 3.407571 0 0.756842 4.283947 0 1.261855 5.162458 0 1.72725 6.030645 0 2.158034 6.879168 0 2.555651 7.701364 0 2.925348 8.489394 0 3.269025 9.237688 0 3.587373 9.941802 0 3.88282 10.59709 0 4.154689 11.20091 0 4.402944 11.75096 0 4.628276 12.24532 0 4.829769 12.68352 0 5.007336 13.06553 0 5.160839 13.39219 0 5.290396 13.66512 0 5.396593 13.88672 0 5.48034 14.06026 0 5.508382 14.20141 0 5.475882 14.29664 0 5.435736 14.34569 0 5.396331 14.37427 0 5.357483 14.39145 0 5.303147 14.40161 0 5.216499 14.38753 0 5.117986 14.31011 0 5.032222 14.16556 0 4.925291 13.97975 0 4.794051 13.75116 0 4.637591 13.47789 0 4.455349 13.15873 0 4.247068 12.7931 0 4.012614 12.38115 0 3.752281 11.92351 0 3.466158 11.42161 0 3.154839 10.87721 0 2.81796 10.29324 0 2.456284 9.672514 0 2.069313 9.019089 0 1.656311 8.337673 0 1.217755 7.632718 0 0.750941 6.911134 0 0.255517 6.178895 0 −0.26979 5.44314 0 −0.82937 4.714024 0 −1.424 4.000077 0 −2.05576 3.312094 0 −2.7284 2.664354 0 −3.44364 2.072365 0 −4.19962 1.551693 0 −4.99264 1.121032 0 −5.81387 0.799094 0 −6.6476 0.600091 0 −7.47273 0.528604 0 −8.26383 0.57749 0 −9.00471 0.727382 0 −9.68242 0.952623 0 −10.2932 1.226222 0 −10.8391 1.523592 0 −11.3247 1.82492 0 −11.7556 2.115391 0 −12.1362 2.384712 0 −12.4723 2.623603 0 −12.7675 2.826534 0 −13.0262 2.987445 0 −13.2513 3.101372 0 −13.4439 3.16303 0 −13.5985 3.172318 0 −13.7064 3.145957 0 −14.3653 3.467146 0.1 −14.4088 3.42697 0.1 −14.4608 3.358311 0.1 −14.5128 3.235995 0.1 −14.5384 3.0525 0.1 −14.5192 2.811468 0.1 −14.4476 2.518112 0.1 −14.319 2.177088 0.1 −14.129 1.792989 0.1 −13.8731 1.371411 0.1 −13.5455 0.919904 0.1 −13.1405 0.448567 0.1 −12.6521 −0.02971 0.1 −12.0755 −0.49902 0.1 −11.4076 −0.94028 0.1 −10.6481 −1.33102 0.1 −9.80073 −1.64535 0.1 −8.8749 −1.8547 0.1 −7.88838 −1.9308 0.1 −6.86599 −1.85043 0.1 −5.84022 −1.60247 0.1 −4.8423 −1.19057 0.1 −3.89674 −0.63231 0.1 −3.01816 0.046965 0.1 −2.21222 0.820325 0.1 −1.47923 1.663479 0.1 −0.81398 2.55372 0.1 −0.20831 3.47118 0.1 0.344415 4.400592 0.1 0.848188 5.330367 0.1 1.311552 6.24799 0.1 1.740076 7.143849 0.1 2.135323 8.011117 0.1 2.502573 8.841929 0.1 2.844232 9.630405 0.1 3.160808 10.37207 0.1 3.454883 11.06215 0.1 3.725676 11.69793 0.1 3.973393 12.27693 0.1 4.198377 12.79735 0.1 4.399872 13.25857 0.1 4.577615 13.66065 0.1 4.731423 14.00447 0.1 4.861368 14.29173 0.1 4.967966 14.52496 0.1 5.052115 14.70757 0.1 5.084682 14.85418 0.1 5.051254 14.95358 0.1 5.008484 15.0041 0.1 4.966473 15.03266 0.1 4.925338 15.04889 0.1 4.868315 15.05658 0.1 4.779184 15.03593 0.1 4.683705 14.94729 0.1 4.597422 14.79506 0.1 4.489209 14.59986 0.1 4.356516 14.35967 0.1 4.198482 14.07246 0.1 4.014606 13.73692 0.1 3.804736 13.3524 0.1 3.568849 12.91898 0.1 3.30737 12.43725 0.1 3.020545 11.90859 0.1 2.708932 11.33488 0.1 2.372423 10.71896 0.1 2.011935 10.06367 0.1 1.626783 9.373232 0.1 1.216556 8.652247 0.1 0.781506 7.905394 0.1 0.318718 7.139673 0.1 −0.1721 6.361072 0.1 −0.69273 5.577017 0.1 −1.24805 4.797806 0.1 −1.83953 4.03242 0.1 −2.47006 3.292111 0.1 −3.14461 2.592 0.1 −3.8664 1.949068 0.1 −4.63553 1.381799 0.1 −5.44954 0.912571 0.1 −6.2997 0.565223 0.1 −7.16808 0.359784 0.1 −8.02785 0.304661 0.1 −8.84923 0.391228 0.1 −9.60934 0.593832 0.1 −10.2958 0.879097 0.1 −10.9067 1.214113 0.1 −11.4464 1.571028 0.1 −11.9221 1.928449 0.1 −12.3408 2.27085 0.1 −12.7089 2.587435 0.1 −13.0331 2.868726 0.1 −13.318 3.109423 0.1 −13.569 3.303327 0.1 −13.7899 3.445509 0.1 −13.9828 3.529724 0.1 −14.1425 3.552365 0.1 −14.2559 3.530021 0.1 −14.8946 3.736443 0.2 −14.9399 3.695126 0.2 −14.9941 3.624457 0.2 −15.0496 3.498643 0.2 −15.0807 3.309541 0.2 −15.0697 3.059558 0.2 −15.008 2.753339 0.2 −14.8901 2.395176 0.2 −14.711 1.989623 0.2 −14.4656 1.542266 0.2 −14.1479 1.060505 0.2 −13.7515 0.554201 0.2 −13.2701 0.036112 0.2 −12.6976 −0.47755 0.2 −12.0293 −0.96654 0.2 −11.2627 −1.40596 0.2 −10.3998 −1.7659 0.2 −9.44941 −2.01251 0.2 −8.42979 −2.11159 0.2 −7.37224 −2.03598 0.2 −6.31234 −1.77391 0.2 −5.28614 −1.33254 0.2 −4.31984 −0.73484 0.2 −3.42748 −0.01084 0.2 −2.61348 0.809601 0.2 −1.87624 1.699956 0.2 −1.20908 2.636478 0.2 −0.60294 3.598949 0.2 −0.05091 4.572065 0.2 0.451635 5.543913 0.2 0.913421 6.501993 0.2 1.340346 7.436545 0.2 1.734074 8.340599 0.2 2.099801 9.206322 0.2 2.440295 10.02759 0.2 2.755905 10.79991 0.2 3.049302 11.5184 0.2 3.319628 12.18028 0.2 3.567233 12.78297 0.2 3.792204 13.3247 0.2 3.99391 13.80479 0.2 4.171957 14.22332 0.2 4.326127 14.58121 0.2 4.456465 14.88023 0.2 4.563437 15.12302 0.2 4.647923 15.31311 0.2 4.684316 15.46419 0.2 4.650292 15.56708 0.2 4.60538 15.61882 0.2 4.561246 15.64737 0.2 4.518269 15.6628 0.2 4.4591 15.66848 0.2 4.368199 15.64256 0.2 4.275231 15.5453 0.2 4.188397 15.38679 0.2 4.078976 15.18388 0.2 3.94489 14.93417 0.2 3.785313 14.63554 0.2 3.599792 14.28659 0.2 3.388249 13.88662 0.2 3.150771 13.43565 0.2 2.887832 12.93427 0.2 2.599869 12.38381 0.2 2.28738 11.78625 0.2 1.950493 11.14435 0.2 1.590231 10.461 0.2 1.205786 9.740568 0.2 0.797078 8.987538 0.2 0.364117 8.206805 0.2 −0.09603 7.405383 0.2 −0.58363 6.58929 0.2 −1.10077 5.766137 0.2 −1.65267 4.946261 0.2 −2.24131 4.138949 0.2 −2.87005 3.355622 0.2 −3.5449 2.611936 0.2 −4.27038 1.925876 0.2 −5.04835 1.318035 0.2 −5.87758 0.813453 0.2 −6.74993 0.440254 0.2 −7.64563 0.22331 0.2 −8.53328 0.174066 0.2 −9.38044 0.282369 0.2 −10.1578 0.517648 0.2 −10.854 0.841052 0.2 −11.4682 1.215801 0.2 −12.0064 1.611915 0.2 −12.4772 2.006892 0.2 −12.8889 2.384588 0.2 −13.2489 2.733572 0.2 −13.5653 3.044231 0.2 −13.8429 3.311256 0.2 −14.0884 3.528134 0.2 −14.3064 3.689843 0.2 −14.4997 3.789326 0.2 −14.6633 3.820561 0.2 −14.7811 3.800373 0.2 −15.3978 3.889359 0.3 −15.4442 3.846981 0.3 −15.5001 3.774652 0.3 −15.5582 3.64615 0.3 −15.5933 3.45279 0.3 −15.5877 3.196209 0.3 −15.5318 2.880775 0.3 −15.4193 2.510775 0.3 −15.245 2.090843 0.3 −15.0033 1.626685 0.3 −14.6881 1.125768 0.3 −14.2927 0.597971 0.3 −13.8105 0.056191 0.3 −13.2347 −0.48294 0.3 −12.5595 −0.99817 0.3 −11.7816 −1.46279 0.3 −10.9019 −1.84408 0.3 −9.92921 −2.10456 0.3 −8.8831 −2.20619 0.3 −7.79973 −2.11972 0.3 −6.71697 −1.83389 0.3 −5.6741 −1.35938 0.3 −4.6975 −0.72349 0.3 −3.79994 0.040392 0.3 −2.98443 0.900631 0.3 −2.24762 1.829481 0.3 −1.58174 2.802812 0.3 −0.97714 3.800457 0.3 −0.42687 4.807328 0.3 0.07411 5.811377 0.3 0.534488 6.800235 0.3 0.960239 7.76413 0.3 1.353114 8.695998 0.3 1.718092 9.588067 0.3 2.058146 10.43408 0.3 2.3735 11.22951 0.3 2.666819 11.96943 0.3 2.937177 12.65102 0.3 3.185017 13.27159 0.3 3.41023 13.82944 0.3 3.612277 14.3238 0.3 3.790679 14.7548 0.3 3.945196 15.12338 0.3 4.075869 15.43134 0.3 4.183142 15.6814 0.3 4.267873 15.87719 0.3 4.307333 16.03161 0.3 4.272936 16.13721 0.3 4.226394 16.18988 0.3 4.180649 16.21839 0.3 4.136296 16.23321 0.3 4.075552 16.23735 0.3 3.983465 16.20758 0.3 3.892431 16.10407 0.3 3.805122 15.94085 0.3 3.694667 15.7322 0.3 3.559374 15.47539 0.3 3.398432 15.16823 0.3 3.211416 14.8093 0.3 2.998302 14.39783 0.3 2.759245 13.93381 0.3 2.494764 13.41785 0.3 2.205421 12.85123 0.3 1.891671 12.23601 0.3 1.553864 11.57489 0.3 1.193075 10.87082 0.3 0.808442 10.12823 0.3 0.400188 9.351517 0.3 −0.03187 8.545755 0.3 −0.49052 7.717892 0.3 −0.97601 6.873974 0.3 −1.49056 6.021674 0.3 −2.0394 5.171221 0.3 −2.62496 4.332082 0.3 −3.25067 3.515561 0.3 −3.92313 2.737366 0.3 −4.64786 2.015886 0.3 −5.42813 1.37277 0.3 −6.26386 0.834309 0.3 −7.1482 0.431338 0.3 −8.06158 0.192437 0.3 −8.96938 0.132263 0.3 −9.83776 0.241039 0.3 −10.6326 0.485886 0.3 −11.3419 0.824645 0.3 −11.9649 1.217921 0.3 −12.5082 1.634073 0.3 −12.9809 2.049519 0.3 −13.3921 2.447356 0.3 −13.7502 2.815421 0.3 −14.0639 3.143718 0.3 −14.3387 3.426809 0.3 −14.5821 3.657721 0.3 −14.7994 3.831153 0.3 −14.9941 3.939362 0.3 −15.1608 3.974797 0.3 −15.2816 3.954905 0.3 −15.8748 3.925899 0.4 −15.9218 3.882553 0.4 −15.9785 3.808946 0.4 −16.0384 3.678599 0.4 −16.076 3.482335 0.4 −16.073 3.221367 0.4 −16.0186 2.900173 0.4 −15.9061 2.523331 0.4 −15.7303 2.095712 0.4 −15.4853 1.623288 0.4 −15.165 1.113789 0.4 −14.7625 0.577374 0.4 −14.2711 0.02733 0.4 −13.6839 −0.51909 0.4 −12.9948 −1.03967 0.4 −12.2002 −1.50621 0.4 −11.3013 −1.88407 0.4 −10.308 −2.13365 0.4 −9.24263 −2.21531 0.4 −8.14338 −2.10012 0.4 −7.05075 −1.77916 0.4 −6.00429 −1.26688 0.4 −5.02893 −0.59377 0.4 −4.13545 0.205088 0.4 −3.3253 1.097657 0.4 −2.59376 2.056006 0.4 −1.93232 3.056365 0.4 −1.33126 4.079093 0.4 −0.78381 5.109526 0.4 −0.28468 6.13564 0.4 0.174499 7.145355 0.4 0.599553 8.129 0.4 0.992268 9.079463 0.4 1.357315 9.989089 0.4 1.697693 10.85157 0.4 2.013542 11.66234 0.4 2.307399 12.41652 0.4 2.5783 13.11121 0.4 2.826749 13.7437 0.4 3.052463 14.3123 0.4 3.254989 14.81622 0.4 3.433797 15.25557 0.4 3.588651 15.63133 0.4 3.719603 15.94531 0.4 3.827093 16.20028 0.4 3.911988 16.39992 0.4 3.953794 16.55651 0.4 3.919174 16.664 0.4 3.871528 16.71727 0.4 3.824686 16.74574 0.4 3.779429 16.76011 0.4 3.717669 16.76321 0.4 3.624874 16.73102 0.4 3.535173 16.62344 0.4 3.447496 16.45717 0.4 3.336213 16.24481 0.4 3.199936 15.98344 0.4 3.037844 15.67083 0.4 2.849522 15.30551 0.4 2.634973 14.88671 0.4 2.394396 14.41439 0.4 2.128311 13.88916 0.4 1.837383 13.3123 0.4 1.522013 12.6859 0.4 1.182737 12.01264 0.4 0.820647 11.29547 0.4 0.434909 10.53888 0.4 0.025991 9.747191 0.4 −0.40639 8.925568 0.4 −0.86477 8.080833 0.4 −1.34936 7.219055 0.4 −1.86232 6.347848 0.4 −2.40857 5.477166 0.4 −2.99082 4.616529 0.4 −3.61233 3.776895 0.4 −4.27966 2.973479 0.4 −4.99889 2.224408 0.4 −5.77427 1.55129 0.4 −6.60673 0.980157 0.4 −7.49126 0.542602 0.4 −8.41037 0.269442 0.4 −9.3299 0.17886 0.4 −10.2141 0.264133 0.4 −11.0272 0.49347 0.4 −11.7539 0.823779 0.4 −12.3922 1.214103 0.4 −12.948 1.631357 0.4 −13.4302 2.050745 0.4 −13.8482 2.454302 0.4 −14.2109 2.828989 0.4 −14.5277 3.164075 0.4 −14.8044 3.453763 0.4 −15.0494 3.690544 0.4 −15.2685 3.868576 0.4 −15.4657 3.979511 0.4 −15.635 4.015007 0.4 −15.7575 3.993587 0.4 −16.3258 3.846056 0.5 −16.3728 3.801841 0.5 −16.4296 3.72733 0.5 −16.4902 3.595962 0.5 −16.5289 3.398096 0.5 −16.5255 3.134843 0.5 −16.4683 2.811232 0.5 −16.3504 2.432435 0.5 −16.1667 2.003742 0.5 −15.9113 1.531549 0.5 −15.578 1.024055 0.5 −15.1603 0.491955 0.5 −14.6513 −0.0508 0.5 −14.0445 −0.58613 0.5 −13.3343 −1.0908 0.5 −12.5179 −1.53547 0.5 −11.5982 −1.88444 0.5 −10.5872 −2.09763 0.5 −9.51052 −2.13631 0.5 −8.40669 −1.97463 0.5 −7.31814 −1.60784 0.5 −6.28152 −1.05415 0.5 −5.31874 −0.34565 0.5 −4.43812 0.482689 0.5 −3.63967 1.399745 0.5 −2.9176 2.3784 0.5 −2.26327 3.39594 0.5 −1.66734 4.433689 0.5 −1.12341 5.477541 0.5 −0.62613 6.515665 0.5 −0.16773 7.536414 0.5 0.257294 8.530306 0.5 0.650701 9.490244 0.5 1.016755 10.40873 0.5 1.358337 11.27949 0.5 1.675521 12.09793 0.5 1.970606 12.85925 0.5 2.242633 13.56053 0.5 2.492132 14.19902 0.5 2.71866 14.77307 0.5 2.921853 15.28185 0.5 3.101163 15.7255 0.5 3.256378 16.10496 0.5 3.38758 16.42207 0.5 3.495234 16.6796 0.5 3.580228 16.88127 0.5 3.623722 17.03887 0.5 3.589011 17.14742 0.5 3.54078 17.20098 0.5 3.493351 17.2294 0.5 3.447644 17.24352 0.5 3.385396 17.24607 0.5 3.292287 17.21283 0.5 3.203319 17.10316 0.5 3.115367 16.93546 0.5 3.003449 16.72145 0.5 2.866379 16.45804 0.5 2.703325 16.14301 0.5 2.513854 15.77488 0.5 2.297964 15.35287 0.5 2.055868 14.87695 0.5 1.788057 14.34773 0.5 1.495262 13.76649 0.5 1.177844 13.13534 0.5 0.836441 12.45694 0.5 0.47217 11.73425 0.5 0.084274 10.97174 0.5 −0.32659 10.17368 0.5 −0.76072 9.345253 0.5 −1.22029 8.493119 0.5 −1.70546 7.623325 0.5 −2.21816 6.74335 0.5 −2.76272 5.862703 0.5 −3.34194 4.990832 0.5 −3.95866 4.138193 0.5 −4.61884 3.319031 0.5 −5.32857 2.550588 0.5 −6.0926 1.853478 0.5 −6.91253 1.252092 0.5 −7.78535 0.776703 0.5 −8.69698 0.458419 0.5 −9.61825 0.318598 0.5 −10.5115 0.35597 0.5 −11.3418 0.543587 0.5 −12.0894 0.840308 0.5 −12.7491 1.205009 0.5 −13.3247 1.603556 0.5 −13.8241 2.009834 0.5 −14.2564 2.404468 0.5 −14.6304 2.773298 0.5 −14.9561 3.104512 0.5 −15.2398 3.391528 0.5 −15.4902 3.626257 0.5 −15.7138 3.801973 0.5 −15.9146 3.909745 0.5 −16.0861 3.941166 0.5 −16.2088 3.916395 0.5 −16.7506 3.64984 0.6 −16.797 3.604843 0.6 −16.8533 3.52977 0.6 −16.9138 3.398135 0.6 −16.9519 3.199867 0.6 −16.9452 2.936329 0.6 −16.8808 2.613616 0.6 −16.752 2.237844 0.6 −16.5539 1.814909 0.6 −16.2809 1.351799 0.6 −15.927 0.857412 0.6 −15.486 0.343258 0.6 −14.9514 −0.17576 0.6 −14.3175 −0.68052 0.6 −13.58 −1.14686 0.6 −12.7384 −1.54477 0.6 −11.7981 −1.83872 0.6 −10.7751 −1.99026 0.6 −9.69655 −1.96445 0.6 −8.6013 −1.74102 0.6 −7.53083 −1.32059 0.6 −6.51654 −0.72424 0.6 −5.57625 0.016184 0.6 −4.71579 0.8676 0.6 −3.93394 1.800899 0.6 −3.22437 2.790618 0.6 −2.57886 3.815678 0.6 −1.98895 4.858694 0.6 −1.44861 5.906186 0.6 −0.95267 6.946666 0.6 −0.49425 7.96906 0.6 −0.06826 8.964121 0.6 0.326954 9.924834 0.6 0.695157 10.8439 0.6 1.03901 11.71512 0.6 1.358518 12.53393 0.6 1.655657 13.29563 0.6 1.929514 13.99729 0.6 2.180612 14.63614 0.6 2.40836 15.2106 0.6 2.612498 15.71981 0.6 2.792483 16.16388 0.6 2.948155 16.54375 0.6 3.079641 16.86124 0.6 3.187449 17.11912 0.6 3.272525 17.32108 0.6 3.317135 17.47857 0.6 3.28249 17.58742 0.6 3.234185 17.641 0.6 3.186653 17.66939 0.6 3.140916 17.68343 0.6 3.078656 17.68591 0.6 2.985561 17.65286 0.6 2.896719 17.54288 0.6 2.808538 17.37529 0.6 2.696107 17.16153 0.6 2.558361 16.89845 0.6 2.394432 16.58384 0.6 2.203861 16.21625 0.6 1.986602 15.7949 0.6 1.742857 15.31977 0.6 1.473049 14.79152 0.6 1.177949 14.21138 0.6 0.85787 13.58152 0.6 0.513494 12.90456 0.6 0.145944 12.18346 0.6 −0.24541 11.42265 0.6 −0.65978 10.62632 0.6 −1.0974 9.799655 0.6 −1.56001 8.949089 0.6 −2.04768 8.080606 0.6 −2.56203 7.201506 0.6 −3.10649 6.320718 0.6 −3.68376 5.447476 0.6 −4.29619 4.591675 0.6 −4.94857 3.766244 0.6 −5.64647 2.987036 0.6 −6.39462 2.272965 0.6 −7.19483 1.645787 0.6 −8.04574 1.132651 0.6 −8.93745 0.762797 0.6 −9.84773 0.559223 0.6 −10.7413 0.526903 0.6 −11.5835 0.64689 0.6 −12.3525 0.883569 0.6 −13.0379 1.198019 0.6 −13.6397 1.556123 0.6 −14.1635 1.930672 0.6 −14.6171 2.300605 0.6 −15.0093 2.650246 0.6 −15.3496 2.966446 0.6 −15.6452 3.241143 0.6 −15.9049 3.465629 0.6 −16.1356 3.631864 0.6 −16.3411 3.730289 0.6 −16.514 3.753283 0.6 −16.6356 3.723333 0.6 −17.1494 3.337239 0.7 −17.1945 3.29152 0.7 −17.2496 3.216163 0.7 −17.3091 3.08489 0.7 −17.3451 2.887255 0.7 −17.3321 2.625338 0.7 −17.2561 2.306892 0.7 −17.1107 1.93942 0.7 −16.8917 1.529586 0.7 −16.5943 1.085148 0.7 −16.2126 0.615957 0.7 −15.7408 0.134647 0.7 −15.1737 −0.34264 0.7 −14.5067 −0.79563 0.7 −13.7383 −1.19951 0.7 −12.871 −1.52462 0.7 −11.9141 −1.73752 0.7 −10.8882 −1.80411 0.7 −9.81923 −1.69619 0.7 −8.74756 −1.40067 0.7 −7.70787 −0.92324 0.7 −6.72601 −0.28616 0.7 −5.81548 0.48091 0.7 −4.98004 1.348166 0.7 −4.21762 2.289287 0.7 −3.52186 3.281087 0.7 −2.88558 4.304541 0.7 −2.30146 5.34373 0.7 −1.76386 6.385803 0.7 −1.26806 7.419765 0.7 −0.80828 8.435216 0.7 −0.37984 9.423158 0.7 0.018674 10.37671 0.7 0.39049 11.28882 0.7 0.737965 12.1534 0.7 1.061004 12.96596 0.7 1.361243 13.72192 0.7 1.637834 14.41834 0.7 1.891252 15.05246 0.7 2.12079 15.62279 0.7 2.326289 16.12839 0.7 2.507254 16.5694 0.7 2.663594 16.94671 0.7 2.795498 17.26212 0.7 2.903545 17.51834 0.7 2.988743 17.71902 0.7 3.033998 17.87542 0.7 2.999669 17.98391 0.7 2.951772 18.03729 0.7 2.90458 18.06569 0.7 2.859174 18.07985 0.7 2.797296 18.08271 0.7 2.70448 18.05088 0.7 2.61517 17.94218 0.7 2.52669 17.77602 0.7 2.413746 17.56415 0.7 2.275287 17.30343 0.7 2.110401 16.9917 0.7 1.918574 16.62753 0.7 1.699691 16.21019 0.7 1.453919 15.73967 0.7 1.181575 15.21669 0.7 0.883431 14.64245 0.7 0.559742 14.01915 0.7 0.21121 13.34938 0.7 −0.16108 12.63612 0.7 −0.55761 11.88373 0.7 −0.97748 11.09631 0.7 −1.42084 10.27901 0.7 −1.88896 9.438021 0.7 −2.3817 8.579204 0.7 −2.9004 7.709687 0.7 −3.44741 6.83772 0.7 −4.02502 5.972158 0.7 −4.6351 5.122358 0.7 −5.28096 4.299744 0.7 −5.96711 3.518441 0.7 −6.69776 2.795219 0.7 −7.47451 2.148663 0.7 −8.29693 1.601697 0.7 −9.15887 1.17993 0.7 −10.044 0.905629 0.7 −10.9264 0.78834 0.7 −11.7709 0.818088 0.7 −12.5565 0.96831 0.7 −13.2675 1.206006 0.7 −13.8989 1.499446 0.7 −14.4522 1.821304 0.7 −14.9332 2.148839 0.7 −15.3494 2.464401 0.7 −15.7096 2.753286 0.7 −16.0219 3.005141 0.7 −16.2944 3.210427 0.7 −16.5347 3.359323 0.7 −16.7457 3.441529 0.7 −16.9188 3.451361 0.7 −17.0376 3.414409 0.7 −17.522 2.908259 0.8 −17.5654 2.861829 0.8 −17.6187 2.786363 0.8 −17.6763 2.655889 0.8 −17.7085 2.459687 0.8 −17.6862 2.201182 0.8 −17.5939 1.890542 0.8 −17.4263 1.53715 0.8 −17.1805 1.148552 0.8 −16.8521 0.733438 0.8 −16.4363 0.302865 0.8 −15.9282 −0.12912 0.8 −15.3238 −0.54494 0.8 −14.6213 −0.92331 0.8 −13.8225 −1.23953 0.8 −12.9341 −1.46596 0.8 −11.9711 −1.5739 0.8 −10.9531 −1.53661 0.8 −9.90953 −1.33481 0.8 −8.87423 −0.96278 0.8 −7.87536 −0.42971 0.8 −6.93252 0.243189 0.8 −6.05559 1.030121 0.8 −5.24692 1.905489 0.8 −4.50409 2.846223 0.8 −3.82132 3.83178 0.8 −3.19293 4.845438 0.8 −2.61295 5.87279 0.8 −2.07603 6.901536 0.8 −1.57817 7.92131 0.8 −1.11489 8.922462 0.8 −0.68184 9.896196 0.8 −0.27792 10.83584 0.8 0.099455 11.73461 0.8 0.452357 12.58655 0.8 0.780517 13.38728 0.8 1.085253 14.13234 0.8 1.365821 14.81879 0.8 1.622555 15.44395 0.8 1.854714 16.00633 0.8 2.062237 16.50499 0.8 2.244699 16.94004 0.8 2.402101 17.31232 0.8 2.534716 17.62358 0.8 2.643216 17.87647 0.8 2.728688 18.07462 0.8 2.77426 18.22912 0.8 2.740649 18.33672 0.8 2.693602 18.38981 0.8 2.647141 18.41832 0.8 2.602338 18.43281 0.8 2.541115 18.43642 0.8 2.448738 18.40657 0.8 2.35838 18.30045 0.8 2.269364 18.13672 0.8 2.155694 17.92796 0.8 2.016233 17.67114 0.8 1.850011 17.36413 0.8 1.656427 17.00556 0.8 1.435275 16.59476 0.8 1.18666 16.13176 0.8 0.910777 15.61731 0.8 0.608305 15.05268 0.8 0.279522 14.44001 0.8 −0.07497 13.78195 0.8 −0.45409 13.08143 0.8 −0.85825 12.34279 0.8 −1.28638 11.57002 0.8 −1.73859 10.76824 0.8 −2.21563 9.943392 0.8 −2.71708 9.101137 0.8 −3.24405 8.248494 0.8 −3.79771 7.392973 0.8 −4.37964 6.542891 0.8 −4.99127 5.707139 0.8 −5.63428 4.895597 0.8 −6.31173 4.120479 0.8 −7.02682 3.396212 0.8 −7.78043 2.738025 0.8 −8.57226 2.164183 0.8 −9.39867 1.695467 0.8 −10.2489 1.351007 0.8 −11.1038 1.141851 0.8 −11.9389 1.065576 0.8 −12.7283 1.106618 0.8 −13.4583 1.241812 0.8 −14.1168 1.44532 0.8 −14.7007 1.691741 0.8 −15.212 1.957289 0.8 −15.6561 2.222112 0.8 −16.0403 2.469839 0.8 −16.3728 2.687101 0.8 −16.6611 2.863082 0.8 −16.9125 2.985708 0.8 −17.1291 3.043855 0.8 −17.3005 3.03543 0.8 −17.4149 2.989688 0.8 −17.8685 2.362904 0.9 −17.9097 2.315698 0.9 −17.9607 2.240147 0.9 −18.0155 2.11062 0.9 −18.0422 1.916343 0.9 −18.0069 1.662919 0.9 −17.8938 1.363976 0.9 −17.6988 1.031158 0.9 −17.4207 0.672906 0.9 −17.0565 0.298916 0.9 −16.6025 −0.07839 0.9 −16.0553 −0.44343 0.9 −15.4134 −0.77728 0.9 −14.6783 −1.05826 0.9 −13.8561 −1.26313 0.9 −12.9588 −1.3684 0.9 −12.0025 −1.35278 0.9 −11.0091 −1.19917 0.9 −10.0058 −0.89806 0.9 −9.01765 −0.45006 0.9 −8.06597 0.134164 0.9 −7.16477 0.836394 0.9 −6.32143 1.63597 0.9 −5.53784 2.512069 0.9 −4.81185 3.445112 0.9 −4.13872 4.417333 0.9 −3.51472 5.414431 0.9 −2.9354 6.423516 0.9 −2.39556 7.432686 0.9 −1.8921 8.432281 0.9 −1.42203 9.413486 0.9 −0.98119 10.36762 0.9 −0.56889 11.28823 0.9 −0.18323 12.16887 0.9 0.177612 13.0037 0.9 0.513115 13.78847 0.9 0.824319 14.51884 0.9 1.110625 15.19186 0.9 1.372154 15.80494 0.9 1.608196 16.35662 0.9 1.818784 16.84593 0.9 2.003603 17.27293 0.9 2.162758 17.63841 0.9 2.296627 17.94407 0.9 2.406 18.19246 0.9 2.492059 18.3871 0.9 2.537856 18.53923 0.9 2.505569 18.64564 0.9 2.459761 18.69851 0.9 2.414335 18.72726 0.9 2.370282 18.74232 0.9 2.309827 18.74704 0.9 2.217867 18.7196 0.9 2.125915 18.61694 0.9 2.035878 18.45617 0.9 1.920945 18.25118 0.9 1.779799 17.99906 0.9 1.611379 17.69776 0.9 1.414993 17.34598 0.9 1.190306 16.94313 0.9 0.937321 16.48929 0.9 0.656086 15.98529 0.9 0.34718 15.43242 0.9 0.010867 14.83285 0.9 −0.35238 14.18925 0.9 −0.74155 13.5046 0.9 −1.15697 12.78319 0.9 −1.5974 12.0289 0.9 −2.06296 11.24688 0.9 −2.55387 10.44279 0.9 −3.06931 9.62207 0.9 −3.6103 8.791736 0.9 −4.17668 7.958461 0.9 −4.76914 7.129952 0.9 −5.38872 6.314709 0.9 −6.03531 5.52102 0.9 −6.71026 4.759161 0.9 −7.41533 4.041099 0.9 −8.15008 3.378583 0.9 −8.91355 2.785196 0.9 −9.70315 2.276275 0.9 −10.5121 1.866206 0.9 −11.3281 1.564939 0.9 −12.1344 1.375335 0.9 −12.9134 1.29271 0.9 −13.6464 1.305899 0.9 −14.3215 1.397913 0.9 −14.9299 1.547696 0.9 −15.469 1.731807 0.9 −15.9408 1.928602 0.9 −16.3502 2.12025 0.9 −16.7042 2.290118 0.9 −17.0095 2.425563 0.9 −17.2723 2.511813 0.9 −17.4926 2.537261 0.9 −17.6592 2.50552 0.9 −17.7673 2.449311 0.9 −18.1889 1.701256 1 −18.2276 1.653102 1 −18.2757 1.577296 1 −18.3269 1.448468 1 −18.3459 1.256271 1 −18.2936 1.009479 1 −18.1549 0.726805 1 −17.928 0.421881 1 −17.6141 0.103914 1 −17.2117 −0.21654 1 −16.719 −0.5258 1 −16.1351 −0.80709 1 −15.4616 −1.0408 1 −14.704 −1.20599 1 −13.8729 −1.2825 1 −12.9826 −1.25263 1 −12.0518 −1.10371 1 −11.1011 −0.82853 1 −10.1503 −0.42684 1 −9.21742 0.095428 1 −8.31634 0.727159 1 −7.45631 1.453807 1 −6.6434 2.260489 1 −5.88014 3.13173 1 −5.16547 4.051739 1 −4.49604 5.00549 1 −3.87037 5.981182 1 −3.28606 6.967656 1 −2.73784 7.953184 1 −2.22342 8.928712 1 −1.74169 9.886454 1 −1.28851 10.81771 1 −0.8635 11.71622 1 −0.46567 12.57595 1 −0.0933 13.3911 1 0.252769 14.15757 1 0.573301 14.87118 1 0.867908 15.5289 1 1.136474 16.12825 1 1.378332 16.6678 1 1.593617 17.14652 1 1.782162 17.56442 1 1.944207 17.92224 1 2.080241 18.22158 1 2.191211 18.4649 1 2.278413 18.6556 1 2.324689 18.8052 1 2.29458 18.91037 1 2.250337 18.96329 1 2.206162 18.99252 1 2.162852 19.00843 1 2.103076 19.01457 1 2.0112 18.98963 1 1.917183 18.89077 1 1.825293 18.73296 1 1.708106 18.53168 1 1.564022 18.2842 1 1.391876 17.98857 1 1.190843 17.64359 1 0.960429 17.24875 1 0.700521 16.8042 1 0.410971 16.3109 1 0.092234 15.7702 1 −0.2555 15.1843 1 −0.63188 14.55598 1 −1.036 13.88826 1 −1.46815 13.18545 1 −1.92688 12.45133 1 −2.41245 11.6912 1 −2.92437 10.91036 1 −3.46149 10.11408 1 −4.02482 9.309509 1 −4.61249 8.502206 1 −5.22454 7.699445 1 −5.86135 6.90928 1 −6.5206 6.138033 1 −7.20199 5.394145 1 −7.9051 4.6867 1 −8.62738 4.023471 1 −9.36639 3.413188 1 −10.1195 2.865595 1 −10.8825 2.390364 1 −11.6494 1.996738 1 −12.4119 1.692767 1 −13.1589 1.483554 1 −13.8768 1.368521 1 −14.5515 1.339107 1 −15.171 1.379366 1 −15.7286 1.467629 1 −16.2221 1.581982 1 −16.6533 1.703509 1 −17.0266 1.813483 1 −17.3469 1.897052 1 −17.6183 1.936283 1 −17.8374 1.920737 1 −17.995 1.861716 1 −18.0946 1.793589 1 −18.2519 1.555058 1.020435 −18.29 1.506681 1.020435 −18.3376 1.43078 1.020435 −18.388 1.302027 1.020435 −18.405 1.110213 1.020435 −18.3488 0.864943 1.020435 −18.2045 0.586115 1.020435 −17.9708 0.28768 1.020435 −17.6495 −0.02105 1.020435 −17.2395 −0.32939 1.020435 −16.7395 −0.62347 1.020435 −16.149 −0.88634 1.020435 −15.4704 −1.09843 1.020435 −14.7101 −1.23908 1.020435 −13.879 −1.28891 1.020435 −12.9927 −1.23142 1.020435 −12.0694 −1.05555 1.020435 −11.1288 −0.75589 1.020435 −10.19 −0.33384 1.020435 −9.26907 0.203585 1.020435 −8.37855 0.845063 1.020435 −7.52671 1.57664 1.020435 −6.71941 2.384573 1.020435 −5.95948 3.254574 1.020435 −5.24624 4.171683 1.020435 −4.57661 5.121423 1.020435 −3.9496 6.092504 1.020435 −3.36335 7.074193 1.020435 −2.81258 8.054772 1.020435 −2.29508 9.025263 1.020435 −1.81021 9.978138 1.020435 −1.3538 10.90467 1.020435 −0.92554 11.79862 1.020435 −0.52462 12.65407 1.020435 −0.14935 13.46519 1.020435 0.199383 14.22793 1.020435 0.522269 14.93814 1.020435 0.818967 15.59275 1.020435 1.089323 16.18933 1.020435 1.332679 16.72643 1.020435 1.549184 17.20304 1.020435 1.738715 17.61912 1.020435 1.901533 17.9754 1.020435 2.038162 18.27349 1.020435 2.149579 18.5158 1.020435 2.237107 18.70572 1.020435 2.283556 18.85484 1.020435 2.254015 18.95978 1.020435 2.210118 19.01276 1.020435 2.166206 19.04214 1.020435 2.12303 19.05825 1.020435 2.063358 19.06472 1.020435 1.97142 19.04027 1.020435 1.876943 18.94215 1.020435 1.784536 18.78487 1.020435 1.666723 18.58424 1.020435 1.521832 18.3376 1.020435 1.34867 18.043 1.020435 1.146382 17.69925 1.020435 0.914444 17.30589 1.020435 0.652706 16.86307 1.020435 0.360977 16.37178 1.020435 0.039687 15.83339 1.020435 −0.31099 15.25012 1.020435 −0.69074 14.62477 1.020435 −1.09869 13.96037 1.020435 −1.5351 13.26126 1.020435 −1.99849 12.53119 1.020435 −2.48916 11.77552 1.020435 −3.00644 10.99944 1.020435 −3.54916 10.20824 1.020435 −4.11829 9.409075 1.020435 −4.71155 8.607225 1.020435 −5.32892 7.809941 1.020435 −5.97058 7.025136 1.020435 −6.63367 6.258638 1.020435 −7.31755 5.518534 1.020435 −8.02125 4.813194 1.020435 −8.74162 4.149455 1.020435 −9.47583 3.534992 1.020435 −10.2211 2.978479 1.020435 −10.9739 2.488961 1.020435 −11.7297 2.076104 1.020435 −12.4818 1.749333 1.020435 −13.2206 1.515484 1.020435 −13.9332 1.375515 1.020435 −14.6054 1.322091 1.020435 −15.2254 1.340097 1.020435 −15.7851 1.408315 1.020435 −16.2818 1.505079 1.020435 −16.7165 1.611215 1.020435 −17.0931 1.707946 1.020435 −17.4159 1.779955 1.020435 −17.6882 1.808618 1.020435 −17.9064 1.784099 1.020435 −18.0614 1.719368 1.020435 −18.1591 1.648713 1.020435 −18.3149 1.408912 1.040863 −18.3525 1.360299 1.040863 −18.3995 1.284282 1.040863 −18.449 1.155565 1.040863 −18.4641 0.964084 1.040863 −18.404 0.720339 1.040863 −18.2539 0.445418 1.040863 −18.0136 0.153545 1.040863 −17.685 −0.14589 1.040863 −17.2678 −0.4421 1.040863 −16.7607 −0.72104 1.040863 −16.1642 −0.96567 1.040863 −15.4811 −1.15646 1.040863 −14.7186 −1.27312 1.040863 −13.8883 −1.29703 1.040863 −13.0065 −1.21282 1.040863 −12.0908 −1.01091 1.040863 −11.1607 −0.6875 1.040863 −10.2338 −0.24553 1.040863 −9.3248 0.306937 1.040863 −8.44473 0.958257 1.040863 −7.60089 1.694945 1.040863 −6.79894 2.504315 1.040863 −6.04206 3.373271 1.040863 −5.32994 4.287698 1.040863 −4.65983 5.233648 1.040863 −4.03117 6.200328 1.040863 −3.44273 7.177469 1.040863 −2.88918 8.15335 1.040863 −2.36838 9.119045 1.040863 −1.88017 10.06729 1.040863 −1.42037 10.98934 1.040863 −0.98869 11.87898 1.040863 −0.58454 12.73036 1.040863 −0.20623 13.53767 1.040863 0.145282 14.2969 1.040863 0.470633 15.00389 1.040863 0.769523 15.65559 1.040863 1.041758 16.24957 1.040863 1.286692 16.78438 1.040863 1.504492 17.25899 1.040863 1.695069 17.67337 1.040863 1.858719 18.02823 1.040863 1.995986 18.32515 1.040863 2.107882 18.56652 1.040863 2.195765 18.75571 1.040863 2.24243 18.90438 1.040863 2.213485 19.00914 1.040863 2.169927 19.06221 1.040863 2.126271 19.09173 1.040863 2.083206 19.10807 1.040863 2.023616 19.11485 1.040863 1.931581 19.09085 1.040863 1.836654 18.9934 1.040863 1.74369 18.83659 1.040863 1.625198 18.63655 1.040863 1.479436 18.39065 1.040863 1.30518 18.09696 1.040863 1.101547 17.75432 1.040863 0.867978 17.36229 1.040863 0.604286 16.92103 1.040863 0.310246 16.43158 1.040863 −0.01375 15.8953 1.040863 −0.36754 15.31446 1.040863 −0.75084 14.69186 1.040863 −1.16281 14.03057 1.040863 −1.6037 13.33494 1.040863 −2.07199 12.6087 1.040863 −2.56801 11.85726 1.040863 −3.09091 11.08572 1.040863 −3.63952 10.29939 1.040863 −4.21474 9.505411 1.040863 −4.81389 8.708795 1.040863 −5.43692 7.916791 1.040863 −6.08375 7.137132 1.040863 −6.75097 6.375146 1.040863 −7.43764 5.638551 1.040863 −8.14214 4.934963 1.040863 −8.86072 4.270263 1.040863 −9.59009 3.651069 1.040863 −10.3274 3.085073 1.040863 −11.0698 2.580946 1.040863 −11.8142 2.149035 1.040863 −12.5558 1.800202 1.040863 −13.2862 1.542837 1.040863 −13.9933 1.379145 1.040863 −14.6628 1.302796 1.040863 −15.2825 1.299383 1.040863 −15.8439 1.348115 1.040863 −16.3433 1.42767 1.040863 −16.7811 1.518585 1.040863 −17.1605 1.602158 1.040863 −17.4854 1.662647 1.040863 −17.7585 1.680758 1.040863 −17.9754 1.647372 1.040863 −18.1278 1.577063 1.040863 −18.2236 1.503916 1.040863 −18.2793 1.4492 1.040863

In the above Table 1, the plane Zadim=0 corresponds to the reference plane P0 situated at the base of the profile. This plane P0 is the intersection of the stacking axis AE of the set of vanes with the axisymmetric surface of the hub, i.e. it passes via the intersection between the axis AE and the surface 16A where it is assembled with the inner shroud 16. The plane Zadim=1 corresponds to the reference plane P1 situated at the top of the profile. This plane P1 is the intersection between the axis AE and the axisymmetric surface of the casing, i.e. it passes via the intersection of the axis AE with the surface 12A where it is assembled with the outer ring 12.

By convention, the stacking axis AE of the set of vanes, for the vane of a nozzle, is the axis that extends in the radial direction Z and passes through the middle of the inter-vane throat. For a given vane, the inter-vane throat is the location where the distance between the trailing edge of the vane in question and the suction side of the preceding vane in the direction of rotation of the turbine is at a minimum.

This is the axis on which the various X,Y sections of the set of vanes are “stacked” when designing the set of vanes.

As mentioned at the beginning of the present description, the aerodynamic profile of the invention is substantially identical to the nominal profile defined in the above table, i.e. it departs from said nominal profile by very little at most, being defined in particular within an envelope of ±1 mm in a direction normal to the surface of the nominal profile, and/or having X,Y coordinates lying within a range of ±5% relative to the coordinates X,Y of the nominal profile.

The coordinate Zadim is non-dimensional, i.e. for a point P, situated at a distance D from the plane P0 (where D is measured along the axis Z), the value of Zadim is D/H, where H represents the total height of the profile as measured between the planes P0 and P1. Naturally, in the Table 1 above, by multiplying the coordinate Zadim by the height H, it is possible to obtain the complete coordinates of the vane.

By way of example, the total height H is preferably about 130 mm±10%. By way of example, the plane P0 is situated at a distance of about 395 mm±10% from the axis A.

In the above table, the profile is characterized by fifteen section planes at constant Zadim coordinates, for which the coordinates X and Y are specified. These fifteen section planes include eleven section planes located between Zadim=0 and Zadim=1, and thus arranged in the fluid stream, these section planes being distributed from P0 to P1. The fifteen section planes also include four section planes which are substantially outside the fluid stream (two section planes with Zadim<0, and two section planes with Zadim>1), these four section planes being provided for ensuring geometric continuity of the stacking, close to the head and the foot, respectively. In each section plane at constant coordinate Zadim, the section of the profile is given by a continuous and smooth curve, interconnecting all of the points (X,Y). In each section plane, the profile is interpolated so as to generate a uniform profile. Between the section planes, the profile is to be interpolated so as to generate a complete vane, as homogeneous as possible.

It is advantageous for the nozzle that includes the vane of the invention to have 140 to 160 vanes that present aerodynamic profiles as defined above. 

We claim:
 1. An aerodynamic profile for a turbine vane, the aerodynamic profile defining the turbine vane when cold and in a non-coated state, the aerodynamic profile being substantially identical to a nominal profile determined by Cartesian coordinates X,Y,Zadim given in Table 1, in which the coordinate Zadim is a quotient D/H, where D is a distance of a point under consideration from a reference X,Y plane situated at a base of the nominal profile, and H is a height of said profile measured from said reference plane out to an end of the turbine vane, the measurements D and H being taken radially relative to an axial direction of a turbine, while the coordinate X is measured in the axial direction of the turbine.
 2. The aerodynamic profile as claimed in claim 1, wherein said profile is defined within an envelope of ±1 mm in a direction normal to the surface of the nominal profile.
 3. The aerodynamic profile as claimed in claim 1, wherein the coordinates X,Y of said profile lie within a range of ±5% relative to the coordinates X,Y of the nominal profile.
 4. The aerodynamic profile as claimed in claim 1, wherein the turbine vane is a nozzle vane forming a part of a stator of the turbine.
 5. The aerodynamic profile as claimed in claim 4, wherein the turbine vane is a nozzle vane of a fifth stage of the turbine.
 6. The aerodynamic profile as claimed in claim 4, wherein the turbine vane is a vane of a fifth stage nozzle of the turbine, wherein the turbine includes seven stages.
 7. A turbine vane, presenting an aerodynamic profile as claimed in claim
 1. 8. A turbine, including a plurality of turbine vanes presenting aerodynamic profiles as claimed in claim
 1. 9. A turbine as claimed in claim 8, including a nozzle that is stationary in rotation, having 140 to 160 vanes that each present the aerodynamic profile.
 10. A turbine nozzle forming a portion of a turbine stator, wherein all the turbine vanes of the nozzle present an aerodynamic profile as claimed in claim
 1. 11. A turbine nozzle as claimed in claim 10, including 140 to 160 vanes that each present the aerodynamic profile. 